The shapes of spinning superfluid quantum droplets

In an experiment on nanometer-sized superfluid helium droplets at the FERMI free-electron laser, scientists traced in great detail how the droplets’ equilibrium shapes evolve with increasing rotational speed. Three-dimensional information on individual droplet shapes was retrieved using a wide-angle scattering method. In combination with unprecedented statistics, it was possible to show that the resulting deformation of the rotating droplets is the same as for a classical liquid. This unexpected finding substantially adds to the understanding of superfluid quantum droplets.
For more than a hundred years, the shapes of rotating drops have been extensively studied both experimentally and theoretically. In general, the equilibrium shape of a drop is determined by the balance of surface tension and centrifugal force. With increasing angular momentum, drops of a classical liquid undergo a transition from spherical to oblate and prolate shapes, exhibiting dumbbell-like structures just before fission into two separate drops. 
In contrast, a droplet of a superfluid liquid with zero viscosity cannot rotate as a rigid body. Thus, angular momentum will lead to the formation of quantized vortices inside the droplet to store the rotational energy. This unusual behavior is also expected for nanometer-sized droplets of superfluid helium, where the vortices might affect the equilibrium shapes. However, it was impossible to experimentally study the shape and structure of such fragile systems until the advent of short-wavelength free-electron lasers. These novel light sources allow the structure determination of gas phase nanoparticles using coherent diffractive imaging (CDI) methods. The scattered light of individual helium nanodroplets can be recorded and their shapes can be retrieved from the scattering images [cf. pioneering work by Gomez et al., Science 345, 906 (2014)]. When recorded up to large scattering angles, a single diffraction pattern contains three-dimensional information on the particle. This technique was used in an experiment at the FERMI free-electron laser’s Low Density Matter (LDM) end station. The exceptional spectral properties of FERMI allowed the recording of high quality wide-angle scattering images [Figs. 1(a)-1(e)] that were reproduced by calculating diffraction patterns [Figs. 1 (k)-1(o)] for simple model shapes [Figs. 1(f)-1(j)]. A comparison of these model shapes to theoretical models of classically rotating drops showed a very close resemblance: During their evolution from oblate to prolate geometries, superfluid helium nanodroplets exhibit the same shapes as their classical counterparts. Further, geometries of superfluid droplets beyond the classical stability limit as reported in previous studies, where only projections of the droplet shapes could be retrieved, have not been observed.  With this surprising result, an important step forward was taken in exploring the interplay of droplet geometry and superfluidity on the nanometer scale.

Figure 1.  Transformation of helium nanodroplet shapes. Recorded scattering images (a)-(e), simple model shapes (f)-(j), and corresponding calculated diffraction patterns (k)-(o). Reproduced from PRL 121, 255301 (2018).

This research was conducted by the following research team:

Bruno Langbehn,¹ Katharina Sander,² Yevheniy Ovcharenko,^1,3 Christian Peltz,² Andrew Clark,⁴ Marcello Coreno,⁵ Riccardo Cucini,⁶ Marcel Drabbels,⁴ Paola Finetti,⁶ Michele Di Fraia,^6,5 Luca Giannessi,⁶ Cesare Grazioli,⁵ Denys Iablonskyi,⁷ Aaron C. LaForge,⁸ Toshiyuki Nishiyama,⁹ Verónica Oliver Álvarez de Lara,⁴ Paolo Piseri,¹⁰ Oksana Plekan,⁶ Kiyoshi Ueda,⁷ Julian Zimmermann,^1,11 Kevin C. Prince,^6,12 Frank Stienkemeier,⁸ Carlo Callegari,^6,5 Thomas Fennel,^2,11 Daniela Rupp,^1,11 and Thomas Möller¹

¹ Institut für Optik und Atomare Physik, Technische Universität Berlin,Berlin, Germany
² Institut für Physik, Universität Rostock, Rostock, Germany
³ European XFEL GmbH, Schenefeld, Germany
⁴ Laboratory of Molecular Nanodynamics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
⁵ ISM-CNR, Istituto di Struttura della Materia, Trieste, Italy
⁶ Elettra - Sincrotrone Trieste S.C.p.A., Trieste, Italy
⁷ Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai, Japan
⁸ Physikalisches Institut, Universität Freiburg, Freiburg, Germany
⁹ Division of Physics and Astronomy, Graduate School of Science, Kyoto University, Kyoto, Japan
¹⁰ CIMAINA and Dipartimento di Fisica, Università degli Studi di Milano, Milano, Italy
¹¹ Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Berlin, Germany
¹² Department of Chemistry and Biotechnology, Swinburne University of Technology, Victoria, Australia

Contact persons:

Bruno Langbehn, email: bruno.langbehn@physik.tu-berlin.de
Thomas Fennel,email:thomas.fennel@uni-rostock.de
Daniela Rupp, email: daniela.rupp@mbi-berlin.de